Problem 102 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 101 to answer each of the following. a. Is \(\cos \left(30^{\circ}+60^{\circ}\right),\) or \(\cos 90^{\circ},\) equal to \(\cos 30^{\circ}+\cos 60^{\circ} ?\) b. Is \(\cos \left(30^{\circ}+60^{\circ}\right),\) or \(\cos 90^{\circ},\) equal to \(\cos 30^{\circ} \cos 60^{\circ}-\sin 30^{\circ} \sin 60^{\circ} ?\)

Step-by-Step Solution

Answer

a) \(\cos30^{\circ} + \cos60^{\circ}\) does not equal \(\cos(30^{\circ} + 60^{\circ}) = \cos90^{\circ}\). b) \(\cos30^{\circ}\cos60^{\circ} - \sin30^{\circ}\sin60^{\circ}\) equals \(\cos(30^{\circ} + 60^{\circ}) = \cos90^{\circ}\).

Step 1: Calculation of \(\cos(30^{\circ}+60^{\circ})\) and \(\cos90^{\circ}\)

\(\cos(30^{\circ}+60^{\circ}) = \cos90^{\circ}\). Both expressions evaluate to 0.

Step 2: Calculation of \(\cos30^{\circ} + \cos60^{\circ}\)

Evaluating these separately, we find that \(\cos30^{\circ}\) equals \(\frac{\sqrt{3}}{2}\) and \(\cos60^{\circ}\) equals \(\frac{1}{2}\). Their sum equals \(\frac{\sqrt{3}}{2} +\frac{1}{2}\) which is not equal to 0.

Step 3: Calculation of \(\cos30^{\circ}\cos60^{\circ} - \sin30^{\circ}\sin60^{\circ}\)

By using the values of the individual trigonometric functions, we have this expression as \(\frac{\sqrt{3}}{2}*\frac{1}{2}-\frac{1}{2} * \frac{\sqrt{3}}{2}\) which simplifies to 0.

Step 4: Comparing the results

From the above steps we can see that \(\cos30^{\circ} + \cos60^{\circ}\) does not equal to \(\cos(30^{\circ} + 60^{\circ})\) or \(\cos90^{\circ}\) but \(\cos30^{\circ}\cos60^{\circ} - \sin30^{\circ}\sin60^{\circ}\) equals to \(\cos(30^{\circ}+60^{\circ}) = \cos90^{\circ}\).

Key Concepts

Cosine Addition Formula

When dealing with angles, one of the most important tools in trigonometry is the cosine addition formula. It helps us find the cosine of the sum of two angles in a very efficient way.

The formula is expressed as:
\( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).
This shows that to find the cosine of two added angles, you don't just add their individual cosines. Instead, you multiply the cosines of each angle and subtract the product of their sines.

For example, in the exercise above, it asks for \( \cos (30^\circ + 60^\circ) \). Using the formula, this breaks down into:

• \( \cos 30^\circ \cos 60^\circ \) minus \( \sin 30^\circ \sin 60^\circ \).

By solving, you find that both sides result in zero, proving that using this formula is often the correct approach and not merely adding the cosine values directly.

Trigonometric Functions

Trigonometric functions are essential in mathematics for understanding relationships in triangles and wave functions.

The primary trigonometric functions include:

• Cosine (\( \cos \))
• Sine (\( \sin \))
• Tangent (\( \tan \))

In the context of the exercise, cosine and sine are the functions of interest.
For specific angles like 30° and 60°:

• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 60^\circ = \frac{1}{2} \)
• \( \sin 30^\circ = \frac{1}{2} \)
• \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)

It's crucial to know these values as they frequently appear in trigonometry problems. By combining these values according to the angles involved, we can solve various cosine and sine-related equations.

Angle Addition Identities

Angle addition identities are fundamental formulas in trigonometry. They allow calculations involving the trigonometric functions of compound angles, such as the sum or difference of two angles.

These identities include:

• Cosine addition identity: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• Sine addition identity: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

Using these identities is particularly beneficial in simplifying expressions and solving equations that include these compound angles.
In practice, these identities account for the different ways angles can be combined, providing a way to calculate more complex trigonometric expressions based on the simpler ones.

Understanding and applying these identities are vital when tasked with anything beyond basic angles, enabling us to analyze and solve more complex trigonometric problems accurately.